14th International Symposium on Particle Image Velocimetry – ISPIV 2021
August 1–5, 2021

Error propagation dynamics of velocimetry-based
pressure field calculations (2): on the error profile

Matthew Faiella1, Corwin Grant Jeon Macmillan1,
Jared P. Whitehead2∗ and Zhao Pan1†

1 University of Waterloo, Department of Mechanical and Mechatronics Engineering, ON N2L 3G1, Canada
2 Brigham Young University, Mathematics Department, Provo, UT 84602, USA

∗ whitehead@mathematics.byu.edu † zhao.pan@uwaterloo.ca

Abstract
This work investigates the propagation of error in a Velocimetry-based Pressure field reconstruction (V-
Pressure) problem to determine and explain the effects of error profile of the data on the error propagation.
The results discussed are an extension to those found in Pan et al. (2016). We first show how to determine the
upper bound of the error in the pressure field, and that this worst scenario for error in the data field is unique
and depends on the characteristics of the domain. We then show that the error propagation for a V-Pressure
problem is analogous to elastic deformation in, for example, a Euler-Bernoulli beam or Kirchhoff-Love plate
for one- and two-dimensional problems, respectively. Finally, we discuss the difference in error propagation
near Dirichlet and Neumann boundary conditions, and explain the behavior using Green’s function and the
solid mechanics analogy. The methods discussed in this paper will benefit the community in two ways: i)
to give experimentalists intuitive and quantitative insights to design tests that minimize error propagation
for a V-pressure problem, and ii) to create tests with significant error propagation for the benchmarking of
V-Pressure solvers or algorithms. This paper is intended as a summary of recent research conducted by the
authors, whereas the full work has been recently published (Faiella et al., 2021).

1 Introduction
Using Particle Image Velocimetry (PIV) to reconstruct the pressure field in a fluid flow has many promising
applications (for example, see Zhang and Porfiri (2019); Zhang et al. (2020); Deem et al. (2020); Pereira
et al. (2020)). While uncertainty quantification for velocity fields obtained from PIV experiments has been
studied in depth (Wieneke, 2017; Raffel et al., 2018; Sciacchitano, 2019), less research has been conducted
into assessing uncertainty in the calculated pressure field.

de Kat and Van Oudheusden (2012) provides the first analysis on the error associated with sampling
frequency (spatial and temporal) in the context of V-Pressure problems. It was commented that the central
finite difference based pressure Poisson solver acts as a low-pass filter, effectively eliminating the high-
frequency errors (and signals) in the pressure calculation. The ratio of the grid spacing of the numerical
method to the temporal or spatial wave length of the experimental data impacts the frequency response
of the pressure Poisson solver. Specifically, high frequency data (or noise) is filtered resulting in the loss
of high-frequency physics (or low-pass filtering effect). Similarly, low frequency errors (and signals) are
more likely to propagate through the pressure calculation. However, the results are constrained to a ‘local’
analysis of a specific numerical scheme. The effect of the ‘global’ setup of the flow domain, including size,
shape of the domain and configuration of boundary conditions on the frequency response was not covered.

Pan et al. (2016) analytically discussed how the error in the data field propagates to the calculated
pressure field. The upper bounds of the error in the calculated pressure field are derived for different setups.
It showed that the error propagation dynamics of a V-Pressure problem can be significantly affected by the
‘global’ setup of the domain, especially the configuration of the boundary conditions. Under the general
analytical framework, this work indicated that the error propagation is also influenced by the error profile in
the data field. However, this paper did not identify a worst case error profile in the data field that leads to



the maximum overall error in the calculated pressure field. Moreover, how the error in the data field with
different profiles affects the error propagation was not discussed.

In the present work, we show a systematic method for finding the worst case error profile in the data
field which causes the most error in the pressure field to be reached. The procedure for finding this worst
case (or most dangerous) error in the data field demonstrates how the profile of the error in the data (e.g.,
spatial frequency and location of error peaks in the domain) combined with the fundamental configuration
of the domain (i.e. size, dimension, shape, and boundary condition configuration of the domain) affects the
error propagation.

2 Worst case error and the effect of fundamental domain configuration
The error propagation from the data field (ε f ) to the calculated pressure field (εp) for a V-pressure problem
can be modelled via a Poisson’s equation with respect to error (Pan et al., 2016):

ε f = ∇
2
εp. (1)

The power of the error, averaged over the space of the domain, can be measured by the L2 norm; for example,
the error level of the calculated pressure field is

||εp||L2(Ω) =

√∫
ε2

pdΩ

|Ω|
, (2)

where Ω is the domain of the flow field, and |Ω| is the length, area or volume of the domain, for a 1D, 2D, or
3D problem, respectively. To quantitatively assess error propagation from the data field to the pressure field,
the ratio between the error levels pressure field (can be considered as an output) and data field (considered
as an input) can be used. Thus, to find the worst case error for a given domain, we seek the function ε f to
satisfy:

Ar∗ = max
ε f

Ar = max
ε f

||εp||L2(Ω)

||ε f ||L2(Ω)
, (3)

subject to (1) with the appropriate BCs on the domain. Ar is the ratio of the error level in the calculated
pressure field to the error level in the data (i.e. Ar =

||εp||L2(Ω)

||ε f ||L2(Ω)
). We will refer to Ar as the error amplification

ratio, and refer to the ε f function which satisfies (3) as the worst error, denoted ε∗f . The worst error ε∗f can be
thought of as the most dangerous error in the data field that corrupts the pressure field reconstruction most.
We will only be considering cases where ε f is non-zero in order to avoid division by zero, and to reflect the
reality that all experimental data will have some error.

As an example, we will consider the optimization problem shown in (3) with pure Dirichlet boundary
conditions (note that Neumann or mixed boundaries can be treated similarly). Applying the calculus of
variations (Gelfand and Fomin, 1991), we can find that the calculated pressure field will satisfy the Euler-
Lagrange equations:

∇
4
εp =−

1
λ

εp, (4)

with εp = 0 and ∇2εp = 0 on ∂Ω, subject to |Ω|‖∇2εp‖2
L2(Ω) = 1, when accurate Dirichlet boundary condi-

tions are applied.
Equation (4) also appears as the characteristic equation of vibrations of elastic bodies (Timoshenko

et al., 1937), which will be discussed further in §3. As long as the operator remains self-adjoint, as is
the case for the example shown, then there will be a countable number of solutions to the system, each
solution corresponding to a natural frequency βn =

4
√
−λ−1

n , where λn > 0,n = 1,2,3... are the eigenvalues.
To obtain ε∗f , the smallest of these eigenvalues −λ

−1
1 is used. The corresponding eigenfunction will be the

worst error function which yields the highest possible error amplification ratio. Fundamental configurations
of the flow domain, such as the size and shape of the domain, the dimension of the domain, and the type and
configuration of boundary conditions will affect the eigenvalue (see also Pan et al. (2016)), and consequently
the error propagation.



Table 1: Type of boundary conditions (BCs) of the original pressure Poisson equation, the corresponding
BCs of the eigenvalue problem of the worst error and the induced natural boundaries. G, g, H, and h are
functions on the boundary ∂Ω, and n̂ is the unit outward pointing normal on ∂Ω.

Type of BCs
BCs of pressure
Poisson equation

Essential BCs of
eigenvalue problem

Natural BCs of
eigenvalue problem

Dirichlet p = G εp = g ∇2εp = 0
Neumann ∇p · n̂ = H ∇εp · n̂ = h ∇

(
∇2εp

)
· n̂ = 0

The fourth order eigenvalue problem resulting from the fourth order variational problem can be complex
and require long calculations even for simple 1D cases. While this calculation may be unfamiliar to fluid
mechanics researchers, it has been studied extensively in solid mechanics (e.g., Timoshenko et al. (1937)),
taking the form of the Euler-Bernoulli beam problem in 1D, and Kirchoff-Love plate problem in 2D. Tables
of solutions for standard boundary conditions and simple domains can be found in several solid mechanics
textbooks (e.g. Harris and Piersol (2002)), and would be easy to look up for researchers using Table 1.

In a 1D system with two Dirichlet BCs, for example, the solution to the eigenvalue problem (4) is as
follows (see Faiella et al. (2021) for a more detailed derivation):

ε f n =±
√

2sin
(nπ

L
x
)
, n = 1,2,3, . . . (5)

which will yield corresponding error in the calculated pressure field:

εpn =±
√

2
L2

n2π2 sin
(nπ

L
x
)
, n = 1,2,3, . . . (6)

Substituting (5) and (6) into (3), the error amplification ratio is

Ar = β
−2
n =

L2

π2n2 , n = 1,2,3, . . . (7)

Noting the presence of the n2 term in the denominator of (7), the worst error occurs for the first eigen-
value (n = 1 and Ar∗ = L2/π2). Thus, the error amplification ratio is larger for lower values of n, which
corresponds to lower fundamental frequencies in the data error, while Ar→ 0 as n→ ∞. The Poisson equa-
tion acts as a low-pass filter for the data error, allowing lower frequency error to propagate while eliminating
the high frequency errors. Also, the L2 term in the numerator of (7) shows that the error amplification ratio is
increased as the length scale of the domain increases. In general, such a trend also holds for other boundary
condition configurations. For example, for a 1D system equipped with one Dirichlet BC and one Neumann
BC at each end, the error amplification ratio is

Ar =
4L2

π2(2n−1)2 ,n = 1,2,3, . . . (8)

and also possesses low-pass filter characteristics.
This low-pass filtering behaviour is a fundamental property of the Poisson equation, and thus will be

presented regardless of numerical solving scheme chosen. We wish to emphasize that the frequency response
shown in this paper is different than the results obtained when ‘local’ analysis is used. For example, in de Kat
and Van Oudheusden (2012), the amplitude response of a signal or error passing through a Poisson solver is
given by:

TPS(h,λx) =
|p|
| f |

=
|εp|
|ε f |

=
1+ cos

(
π

2h
λx

)
2 sinc

(
2h
λx

) , (9)

where TPS is the transfer function of the Poisson solver, h is the grid spacing, and λx is the spatial wavelength
of the input signal (or error) to the numerical Poisson solver. In (9), as h/λx→ 0, TPS(h,λx)→ 1. This implies
that if the mesh is sufficiently fine, the error propagating through a Poisson solver is expected to be unfiltered.
This expectation holds only when analysing a local region in a domain and is up to the local choice of



numerical scheme and grid spacing. The analysis in the current work (e.g., (7) and (8) for 1D) is a ‘global’
result that holds for all numerical pressure Poisson solvers, and complements the aforementioned ‘local’
analysis. Invoking λx ∼ 1/n, Ar(λx) = Cλ2

x , where C is a constant which is dependant on the fundamental
features of the domain.

3 Analogy to bulking elastic bodies
Interestingly, a parallel can be drawn between the deformation of elastic bodies and the propagation of error
in V-Pressure experiments. We next use Euler-Bernoulli beam theory, as a 1D example, to demonstrate the
intuitive analogy and evaluate the magnitude of the error propagation in V-Pressure. Consider a homoge-
neous beam of length L which undergoes transverse vibrations. The normalized deflection profile of the
beam is governed by the differential equation:

d4Y
dx4 =

d2Y
dt2 , (10)

where Y (x, t) is the beam deflection in the direction that is perpendicular to the x coordinate, and t is time.
By separation of variables (Y (x, t) = X(x)T (t)), where T (t) is a function of t and X(x) is a function of the
spatial x coordinate and hence describes the modes of the vibrating beam, the corresponding eigenvalue
problem of (10) is:

d4X
dx4 = γX , (11)

where γ is the eigenvalue, and k = 4
√

γ (Timoshenko et al., 1937) is the spatial frequencies of the buckled
beam. Different eigenvalues (γn = ω−2

n , n = 1,2,3, ...) correspond to different natural frequencies (ωn) of
the vibrating beam and the eigenfunctions (Xn) describe the modes of the beam deflections. Larger eigen-
values correspond to beam modes with higher spatial frequencies. We also recall the normalized governing
equations of a bending beam:

dX
dx

= Θ, (12)

d2X
dx2 =−M, (13)

d3X
dx3 =−Q, (14)

d4X
dx4 =−W, (15)

where Θ is the slope, M is the bending moment, Q is the shear force in the beam, and W is the transverse
load on the beam, respectively.

One may notice that (11), which is from Euler-Bernoulli beam theory, is the 1D equivalent of (4), which
arises in the error propagation problem of pressure-velocimetry calculations. In addition, (13) describes the
relationship between the second order derivative of the beam deflection and bending moment; and it takes
the form of the Poisson’s equation about error propagation, i.e., (1) in one dimension. We now observe
the analogy between the Euler-Bernoulli problem and the dynamics of V-Pressure error propagation, from
(13) and (1): the buckling of a beam (X) as a result of an applied bending moment (M) is equivalent to the
profile of the error in the pressure field (εp) as a result of the error in the data (ε f ) for the V-Pressure error
propagation problem.

Solving (11) requires four boundary conditions. For example, if both ends of the beam are simply
supported (also called hinged in some textbooks, see figure 1(a) for illustration), the displacement at the
ends are constrained by the kinematic boundary conditions (also called essential boundary conditions):

X(0) = X(L) = 0, (16)

and natural boundary conditions indicating that no bending moments are exerted at the ends (see Harris and
Piersol (2002) and (13)):

X ′′(0) = X ′′(L) = 0. (17)



(a)

(b)(b)

(a) (b)(b)

Figure 1: Bent Euler-Bernoulli beam with different supporting mechanisms: (a) simply supported at both
ends (associated with Dirichlet-Dirichlet BCs for V-Pressure problem) and (b) simply supported at the left
end and supported by a slide at the right end (associated with Dirichlet-Neumann BCs).

We can see that the boundary conditions (16) mimic the essential boundary conditions of the eigenvalue
problem (4) that arise from homogeneous boundary conditions, εp(0) = εp(L) = 0. Similarly, the boundary
conditions (17) mimic the corresponding natural boundary conditions that arise from the calculus of varia-
tions ε′′p(0) = ε′′p(L) = 0. The analogy between beam deflection and V-Pressure error propagation is shown
more clearly in Table 2. This table shows the physical interpretation of different variables/equations in the
beam problem, as well as their equivalent variables/equations in V-Pressure error propagation. The table
also draws the parallel between a homogeneous Neumann boundary in the pressure Poisson equation and a
slide-support in elastic dynamics (illustrated in figure 1(b)). The analogy of the buckling of elastic bodies
will be further reinforced in the following section (§4) in the context of V-Pressure error propagation.

4 Impact of the location of the error
The dynamics of error propagation to the pressure field from the data field is a function that depends on both
the domain’s fundamental features and the location of the error. To demonstrate this, we can use a sharp
peak error function that is constrained to have a finite error power. For example, the following rectangular
peak can be used for 1D analysis:

ε f = δΠ(x− x0) =

{
ξ, |x− x0| ≤ Π

2
0, |x− x0|> Π

2
, (18)

where δΠ(x−x0) is a rectangular pulse function with a small pulse width Π centered at x = x0. The resulting
εp and the corresponding error amplification ratio (Ar) for each x0 ∈ (0,L) then can be evaluated by solv-
ing (1). Ar = Ar(x0) is thus a quantification of how sensitive the pressure solver is to the error at a given
location x0.

0 0.2 0.4 0.6 0.8 1
-0.08

-0.06

-0.04

-0.02

0
(a) (b)

0 0.5 1 1.5 2
0

0.05

0.1

0.15

0.2

0.25

(a) (b)

Figure 2: Error amplification ratio (Ar) and error profile in the pressure field εp when concentrated error
(ε f = δ0.01L(x−x0) is introduced at different locations (x0)). (a) Ar corresponding to the location of concen-
trated error (x0) for domains with different sizes and configurations of boundary condition. (b) Profiles of εp
over a domain with size L = 1, when concentrated error ε f = δΠ(x−x0) is introduced at x0 = 2/3,1/2,1/3.
The solid curves indicate the results on the domain with Dirichlet-Dirichlet BCs, and the dashed curves
indicate the results on the domain with Dirichlet-Neumann BCs, respectively.

Figure 2(a) demonstrates a case study of the error amplification ratio Ar(x0) caused by error concentrated
at different places in a 1D domain with different sizes (L= 1 or 2) and configurations of boundary conditions.



Table 2: Analogy between the beam vibration problem and the error propagation problem raised by the
Velocimetry-based pressure reconstruction.

The concentrated error source is constructed using (18) with width Π = 0.01L and height ξ = 10. The solid
curves, which represent domains with pure Dirichlet boundaries, have the highest value of Ar at the center
of the domain, which is the location which is farthest from any Dirichlet boundary, while Ar vanishes as the
Dirichlet boundaries are approached. The dashed curves, which represent domains with Dirichlet-Neumann
conditions, have the highest value of Ar at the Neumann boundaries, and also have Ar(x0) approach 0 at the
Dirichlet boundaries. This implies that the data near a Dirichlet boundary is less sensitive to error, while
data far from Dirichlet boundaries or close to a Neumann boundary is sensitive to error in the data.

Figure 2(b) shows the reconstructed pressure field, εp, for a concentrated error located at x0 = 1/3,1/2,
and 2/3 for pure Dirichlet (solid lines) and mixed (dashed lines) boundary conditions. The error for the
pure Dirichlet case reaches its highest peak for x0 = 1/2, when the concentrated error is furthest from the
Dirichlet boundaries. For the mixed boundary conditions, the error reaches its highest value for x0 = 2/3,
which is when the concentrated error is closest to the Neumann boundary. The error for the pure Dirichlet
case is much lower than the equivalent error for mixed boundary conditions. Figure 2(b) can be thought of
in terms of the buckling beam example, where x0 is the location of an applied moment on the beam, and
εp is the beam deflection profile. The pure Dirichlet condition would be equivalent to the beam shown in
figure 1(a), while the mixed condition would be equivalent to figure 1(b). The beam shown in figure 1(a)



would deflect the most when the moment was applied to its center, just as the error is maximized for the
pure Dirichlet condition with error at x0 = 1/2. The beam in figure 1(b) would deflect the most for an error
next to the free support, just as the mixed boundary condition has the greatest error for x0 = 2/3. Finally, the
beam in figure 1(b) would deflect much more than the beam in figure 1(a) for an equivalent moment, since
its free end allows more movement and provides less support than the two “fixed” supports.

This effect of Dirichlet and Neumann boundaries on error propagation is also present in 2 dimensions.
The error sensitivity can again be examined using a pulse function of the form:

ε f = δΠ(xxx− xxx000) =

{ 2√
πΠ

, |xxx− xxx000| ≤ Π

2
0, |xxx− xxx000|> Π

2
, (19)

where xxx000 = (x0,y0) is the coordinate of the center of the concentrated error.

(a) (b) log10(Ar)

(a) (b) (c)log10(Ar) log10(Ar)

Figure 3: Error amplification ratio Ar(x0,y0) responding to concentrated error located at xxx000 = (x0,y0) for
domains with (a) pure Dirichlet boundaries and (b) & (c) mixed boundaries, respectively. Dirichlet bound-
aries are marked by green solid lines in (a) & (b), and the green dot in (c). Neumann boundaries are marked
by orange dashed lines.

Figure 3 shows the error sensitivity map on a 1×1 domain for three distinct BC configurations, where
Ar(x0,y0) is a function of the location of the concentrated error source as illustrated. The concentrated error
source is constructed using (19) with a width of Π = 0.02/

√
|Ω|. We again show that error near a Dirichlet

boundary is less sensitive in terms of error propagation, while the error near a Neumann boundary and/or
far away from a Dirichlet boundary is more dangerous. In figure 3(a), Ar is lowest around the edges of the
domain where Dirichlet boundaries are present, while Ar is largest in the center of the domain, far from
the Dirichlet boundaries. In figure 3(b), Ar is again low near the Dirichlet boundaries, but increases along
the Neumann edges. Ar is highest at the center of the Neumann boundary where proximity to a Neumann
boundary is maximized while proximity to a Dirichlet boundary is minimized. Finally, in figure 3(c), Ar is
high everywhere in the domain except for the region near xxx000 = (0,0), where Ar is tamed by the presence of
a Dirichlet boundary.

We now finalize the analogy between the deformation of elastic plates and the V-Pressure error propa-
gation. The behavior of error near boundary conditions can be thought of intuitively using the analogy to a
bending beam in 1D or plate in 2D. An accurate Dirichlet boundary on the V-Pressure domain is analogous
to a simply supported edge on a plate which constrains the deflection to be vanishing at the boundary. We
recall that loads applied next to a simply supported edge cannot deflect the plate much due to the relatively
‘firm’ support nearby. However, as we move away from this firm support, the plate becomes easier to bend.
In contrast, an accurate Neumann boundary on the V-Pressure domain is analogous to a sliding support
which allows any amount of deflection but constrains the local slope to be zero. A moment applied near
this boundary will deform a large segment of the plate; due to the constraint on the slope of the plate at this
boundary, the sliding mechanism will ‘pull’ on other sections of the plate and bring them along with the
loaded section. Long Neumann boundaries are more ‘dangerous’ than short ones, as this effect is carried out
over a longer distance.

These observations about the effect of the distance from an error source to boundaries with different BCs
can also be explained using Green’s function for the Poisson equation. Consider, in 2D, a concentrated error
source in the data field taking the form of the Dirac delta function, ε f = −δ(xxx000), located at xxx000 = (x0,y0).
The error in the pressure field caused by this concentrated error source is the fundamental solution of the



Poisson equation given by the Green’s function. When the distance from the error to a nearby boundary
is much shorter than the distance to any other other boundaries, the local error near the source (and the
nearby boundary) can be approximated by the Green’s function on a half plane by the method of images.
For example, the Green’s function on a half plane with a homogeneous Dirichlet boundary at x = 0 is

εp = G(xxx,xxx000) =
1

2π
ln
[
〈(x,y),(x0,y0)〉
〈(x,y),(−x0,y0)〉

]
, (20)

where 〈(x,y),(x0,y0)〉=
√
(x− x0)2 +(y− y0)2 is the distance from xxx to xxx000 (Tikhonov and Samarskii, 2013).

As the location of the error approaches the boundary (x0→ 0), G(xxx,xxx000)→ 0 and thus, εp vanishes quickly
towards the Dirichlet boundary.

For an error source close to a Neumann boundary, the corresponding Green’s function on the half-
plane is

εp = G(xxx,xxx000) =
1

2π
ln [〈(x,y),(x0,y0)〉〈(x,y),(−x0,y0)〉] . (21)

As the location of the concentrated error source approaches the Neumann boundary (x0→ 0), G(xxx,xxx000)→∞

and εp blows up at the Neumann boundary. This irregular behavior is unique to the Green’s function analysis
and is not physical, as ε f cannot be a strictly Dirac delta function in reality. Despite this singularity, the
mathematical intuition that arises from this analysis shows how the location of the error in the data coupled
with the BCs configuration affects the error propagation.

(a) (b) log10(Ar)

Figure 4: Error amplification ratio Ar(xxx000) corresponding to concentrated error located at xxx000 = (x0,y0) for
domains featuring an airfoil of chord length Lc. Neumann BCs are applied on the surface of the airfoil.
The domains feature (a) Dirichlet BCs on the outer boundaries and (b) mixed BCs on the outer boundaries.
Dirichlet boundaries are marked by green solid lines and Neumann boundaries are marked by orange dashed
lines, respectively.

We further demonstrate the dependence of Ar(xxx000) to the placement of error (xxx000) using a more complex
domain. Figure 4 shows error sensitivity maps for a 3×2 domain with an airfoil, nondimensionalized by the
airfoil chord length, for different BC configurations. Ar(x0,y0) is the error amplification ratio as a function
of the location of concentrated error as illustrated. Figure 4(a) shows that concentrated error near a Dirichlet
BC is tamed by the boundary, as seen by the low value of amplification ratio Ar around the edges of the
domain. Concentrated errors far from Dirichlet boundaries and/or close to a Neumann boundary show more
propagation, and thus a higher value of Ar occurs near the center of its domain. In figure 4(b), we see that
error amplification is highest in the middle of ‘long’ Neumann boundaries such as the left and right edges
of the domain as well as the suction and pressure surface of the airfoil. The leading edge of the airfoil can
be considered a ‘short’ Neumann boundary due to its high curvature, and thus shows less propagation than
these long Neumann boundaries.

Figure 5 shows the reconstructed pressure field for a concentrated error located at different coordinates.
In figure 5(a)-(c), the outer boundaries are Dirichlet, while the airfoil boundaries are Neumann. Figure 5(a)
shows the effect of concentrated error near a Dirichlet boundary; error rapidly dissipates as the Dirichlet
boundary is approached, and does not significantly propagate and contaminate the whole pressure field. In
contrast, figure 5(b) shows a concentrated error near a long Neumann boundary, where the error is amplified
due to its proximity to a Neumann boundary and the center of the domain and contaminates a large area of
the domain. Figure 5(c) also shows a concentrated error near a Neumann boundary, however the pressure



Figure 5: Error in the reconstructed pressure field (εp) caused by concentrated error in the data field located
at different locations (ε f = δ(x0,y0)). The type of BCs of the domain are indicated by green solid lines for
Dirichlet boundaries, and dashed orange lines for Neumann boundaries. The locations of the error (x0,y0)
are marked by the green arrow heads. The legend in each subplot indicates the error amplification ratio
(Ar = ||εp||L2(Ω)/||ε f ||L2(Ω)) corresponding to the the ε f at each location.

field is less contaminated than figure 5(b). This is due to the high curvature of the boundary at this point,
which is considered a ”short” Neumann boundary and thus the error spreads less. In figure 5(d)-(f), the outer
boundaries are mixed boundary conditions, while the airfoil remains as a Neumann boundary. The error
shown in figure 5(d) is now beside a Neumann boundary instead of the Dirichlet boundary in figure 5(a).
As a result, the error propagates much more and the Ar is greater. In figures 5(e) and (f), we see that the
concentrated error propagates more than the error in figure 5(b) and (c). This is because the taming Dirichlet
boundaries are lost and replaced with amplifying Neumann boundaries, so error propagates much more
freely.

5 Conclusion
In the current work, we present a systematic method for finding the worst case error profile in the data field
which causes the most error in the pressure field. This is corresponding to the upper bound of the error in the
calculated pressure field discussed in Pan et al. (2016). The worst case error profile in the data field can be
found by solving an eigenvalue problem derived from an Euler-Lagrange equation maximizing the the error
propagation. The procedure for finding the worst error in the data field shows how the profile of the error
in the data (e.g., spatial frequency and location of error peaks in the domain) coupling with the fundamental
features of the domain (i.e. size, dimension, type of BCs) affects the error propagation. We show that the
Euler-Lagrange equations used to find the worst error function lead to the same eigenvalue problem from
the buckling of an Euler-Bernoulli beam in 1D and a Kirchoff-Love plate in 2D. The data field error can be
thought of as a bending moment applied to an elastic body, with the resulting pressure field being equivalent
to the deformation of the elastic body. Thus, being familiar with the theories of elastic beams and plates
(Timoshenko et al., 1937) can be useful in understanding error propagation in V-Pressure problems in a
more intuitive way.

We point out that error propagation is significantly affected by the location of the error and the error
proximity to Dirichlet and/or Neumann boundaries. This behavior can be explained by finding Green’s
function for the Poisson equation using the method of images. This result can also be explained intuitively
using the solid mechanics analogy. All results in this work represent fundamental properties of the Poisson
equation, which hold regardless of experimental setup and choice of numerical solver. The results and
analogy-based method can help experimentalists studying fluid mechanics to design better tests which avoid
error profiles similar to the worst case, and limit the error in sensitive locations, thus improving overall



accuracy and robustness of the test. In addition, the results can be used to create worst case scenarios and
challenging test cases to benchmark V-Pressure solvers or algorithms.

Finally, we emphasize that the present work studies how error in the data field propagates to the recon-
structed pressure field. How to interpret error in the data field (ε f ) from the error in the velocity field (εuuu)
can be challenging. The error propagation analysis from velocity to data (εuuu → ε f ) involves the temporal
and spatial resolution of velocimetry and specific numerical schemes that evaluate gradients, and thus is out
of the scope of the current analytical research. We will leave this topic for future studies, however, some
relevant results can be found in, for example, McClure and Yarusevych (2019) and Pan et al. (2018).

References
de Kat R and Van Oudheusden B (2012) Instantaneous planar pressure determination from PIV in turbulent

flow. Experiments in fluids 52:1089–1106

Deem EA, Cattafesta III LN, Hemati MS, Zhang H, Rowley C, and Mittal R (2020) Adaptive separation con-
trol of a laminar boundary layer using online dynamic mode decomposition. Journal of Fluid Mechanics
903:A21

Faiella M, Macmillan CGJ, Whitehead JP, and Pan Z (2021) Error propagation dynamics of velocimetry-
based pressure field calculations (2): on the error profile. Measurement Science and Technology
32:084005

Gelfand IM and Fomin SV (1991) Calculus of Variations. Dover

Harris CM and Piersol AG (2002) Harris’ shock and vibration handbook. volume 5. McGraw-Hill New
York

McClure J and Yarusevych S (2019) Generalized framework for piv-based pressure gradient error field
determination and correction. Measurement Science and Technology 30:084005

Pan Z, Whitehead J, Thomson S, and Truscott T (2016) Error propagation dynamics of piv-based pressure
field calculations: How well does the pressure poisson solver perform inherently?. Measurement Science
and Technology 27:084012

Pan Z, Whitehead JP, Richards G, Truscott TT, and Smith BL (2018) Error propagation dynamics of piv-
based pressure field calculation (3): What is the minimum resolvable pressure in a reconstructed field?.
arXiv preprint arXiv:180703958

Pereira LTL, Ragni D, Avallone F, and Scarano F (2020) Pressure fluctuations from large-scale PIV over a
serrated trailing edge. Experiments in Fluids 61:1–17

Raffel M, Willert CE, Scarano F, Kähler CJ, Wereley ST, and Kompenhans J (2018) Particle image ve-
locimetry: a practical guide. Springer

Sciacchitano A (2019) Uncertainty quantification in particle image velocimetry. Measurement Science and
Technology 30:092001

Tikhonov AN and Samarskii AA (2013) Equations of mathematical physics. Courier Corporation

Timoshenko S et al. (1937) Vibration problems in engineering. Technical report

Wieneke B (2017) PIV uncertainty quantification and beyond. Ph.D. thesis. Delft University of Technology

Zhang J, Brindise MC, Rothenberger S, Schnell S, Markl M, Saloner D, Rayz VL, and Vlachos PP (2020)
4d flow mri pressure estimation using velocity measurement-error-based weighted least-squares. IEEE
Transactions on Medical Imaging 39:1668–1680

Zhang P and Porfiri M (2019) A combined digital image correlation/particle image velocimetry study of
water-backed impact. Composite Structures 224:111010


	Introduction
	Worst case error and the effect of fundamental domain configuration
	Analogy to bulking elastic bodies
	Impact of the location of the error
	Conclusion